82-92 Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2

Existence and Nonexistence of Positive Solutions for a
Kirchhoff-type Equation with Inhomogeneous Strong Allee Effect

Ruyun Ma and Guowei Dai
Department of Mathematics, Northwest Normal University, Lanzhou 730070, P.R. China

Email: mary@nwnu.edu.cn, daiguowei@uwnu.edu.cn

Abstract In this paper, we deal with the nonlocal semilinear elliptic equation with inhomogeneous
strong Allee effect {

−M
(∫

Ω
1
2 |∇u|

2 dx
)

∆u = λf(x, u) in Ω,
u = 0 on ∂Ω,

where the nonlocal coefficient M
(∫

Ω
1
2 |∇u|

2 dx
)

is a continuous function of
∫

Ω
1
2 |∇u|

2 dx. By means
of variational approach, we prove that the equation has at least two positive solutions for largeλ under
suitable hypotheses about nonlinearity. We also prove some nonexistence results. In particular, we shall
give a positive answer to the conjecture by Liu, Wang and Shi’s of [1].

Keywords Nonlocal differential equation Variational method Positive solutions Inhomogeneous
strong Allee effect.

1. Introduction

In this paper we study the following problem{
−M

(∫
Ω

1
2 |∇u|

2 dx
)

∆u = λf(x, u) in Ω,
u = 0 on ∂Ω,

(1)

where Ω is a smooth bounded domain in RN for N ≥ 1, the nonlocal coefficient M(t) is a
continuous function of t =

∫
Ω

1
2 |∇u|

2 dx. We shall give a positive answer to a conjecture by
Liu, Wang and Shi’s of [1].

The problem (1) is a generalization of a model introduced by Kirchhoff[2]. More precisely,
Kirchhoff proposed a model given by the equation

ρ
∂2u

∂t2
−

(
ρ0

h
+
E

2L

∫ L

0

∣∣∣∣∂u∂x
∣∣∣∣2 dx

)
∂2u

∂x2
= 0, (2)

where ρ, ρ0, h, E, L are constants, which extends the classical D’Alembert’s wave equation,
by considering the effect of the changing in the length of the string during the vibration. A
distinguishing feature of equation (2) is that the equation contains a nonlocal coefficient ρ0

h +
E
2L

∫ L
0

∣∣∂u
∂x

∣∣2 dx which depends on the average 1
2L

∫ L
0

∣∣∂u
∂x

∣∣2 dx of the kinetic energy 1
2

∣∣∂u
∂x

∣∣2 on
[0, L], and hence the equation is no longer a pointwise identity. The equation{

−
(
a+ b

∫
Ω |∇u|

2 dx
)

∆u = f(x, u) in Ω,
u = 0 on ∂Ω,

(3)

ISSN 1078-6236 International Institute for General Systems Studies, Inc.



Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 83

is related to the stationary analogue of the equation (2). Equation (3) received much atten-
tion only after Lions [3] proposed an abstract framework to the problem. Some important and
interesting results can be found, for example, in [4-15].

In the context of population biology, the nonlinear function f(x, u) ≡ ug(x, u) represents a
density dependent growth if g(x, u) is a function depending on the population density u. While
traditionally g(x, u) is assumed to be declining to reflect the crowding effect of the increasing
population, Allee suggested that physiological and demographic precesses often possess an
optimal density, with the response decreasing as either higher or lower densities. Such growth
pattern is called an Allee effect. If the growth rate per capita is negative when u is small, we
call it a strong Allee effect; if the growth rate per capita is small than the maximum but still
positive for small u, we call it a weak Allee effect (for detail, see [16] or [17]).

Under the special case of equation (3) with a = 1, b = 0 and f(x, u) satisfies inhomoge-
neous strong Allee effect growth pattern, Liu, Wang and Shi[16] prove that the equation{

−∆u = λf(x, u) in Ω,
u = 0 on ∂Ω

(4)

has at least two positive solutions for large λ if
∫ c(x)

0 f(x, s) ds > 0 for x in an open subset
of Ω, where c(x) ∈ C1(Ω) such that f(x, c(x)) = 0(see the assumption of (f2)). They also
prove some nonexistence results. In particular, they conjecture that the nonexistence holds if∫ c(x)

0 f(x, s) ds ≤ 0 for any x ∈ Ω (see Remark 1.7 of [16]).

Motivated by above, we generalize existence and nonexistence results for the semilinear
elliptic equation (4) to the case of nonlocal semilinear elliptic equation (1). More precisely,
if f(x, u) satisfies inhomogeneous strong Allee effect growth pattern and the nonlocal coeffi-
cient M(t) satisfies some suitable conditions, we establish the existence of at least two positive
solutions for the nonlocal problem (1) withλ large enough. We also prove some nonexistence
results for the nonlocal problem (1). In particular, we shall give a positive answer to the con-
jecture by Liu, Wang and Shi. To the best of our knowledge, this is the first paper that discusses
the nonlocal semilinear elliptic equation with inhomogeneous strong Allee effect via variational
method.

We point out the nonlocal coefficient M(t) raises some of the essential difficulties. For
example, the way of proving the geometry condition of Mountain Pass Theorem in [16] can not
be used here because the functional of (1) is notC2 function under our assumptions. In order
to overcome this difficulty, we divided Ω into B1 and B2 by comparing the value of c(x) with
b, then use poincaré inequality to prove it(see Lemma 3.3).

This paper is organized as follows. In Section 2, we present our main resuts and some
necessary preliminary lemmas. In Sections 3, we use variational method and sub-supersolution
method to prove the main results. In Section 4, we prove the conjecture of Liu, Wang and Shi’s
and give some examples which satisfy our hypotheses.

2. Main Resuts and Preliminaries

In this section, we give our main results and some necessary preliminary lemmas which will
be used in the following proof. For simplicity we write X = H1

0 (Ω) with the norm ‖u‖ =



84 Ma:Existence and Nonexistence of Positive Solutions for a Kirchhoff-type . . . . . .(∫ 1
0 |∇u|

2 dx
) 1

2 .

Hereafter, f(x, t) and M(t) are always supposed to verify the following assumptions:

(f1) f(x, u) ∈ C(Ω× R+) and f(x, ·) ∈ C1(R+) for any x ∈ Ω;

(f2) There exist b(x) ∈ C(Ω), c(x) ∈ C1(Ω) such that 0 < b(x) < c(x) and f(x, 0) =
f(x, b(x)) = f(x, c(x)) = 0 for any x ∈ Ω;

(f3) For almost all x ∈ Ω, f(x, s) < 0 for any s ∈ (0, b(x)) ∪ (c(x),+∞) and f(x, s) > 0 for
any s ∈ (b(x), c(x)).

Remark 2.1. Note that the weak maximum principle (Theorem 8.1 of [18]) and strong max-
imum principle (Theorem 8.1 of [18]) also hold for the nonlocal problem (1) because M(t)
satisfies the assumption (M ).

(M ) ∃m0 > 0 such that
M(t) ≥ m0.

Definition 2.1. We say that u ∈ X is a weak solution of (1), if

M

(∫
Ω

1

2
|∇u|2 dx

)∫
Ω
∇u∇ϕdx = λ

∫
Ω
f(x, u)ϕdx

for any ϕ ∈ X.

Define

Φ(u) = M̂

(∫
Ω

1

2
|∇u|2 dx

)
,Ψ(u) =

∫
Ω
F (x, u) dx,

where M̂(t) =
∫ t

0 M(s) ds, F (x, u) =
∫ u

0 f(x, t) dt. We redefine f(x, u), such that f(x, u) ≡
0 when u ∈ (−∞, 0) ∪ (c(x),∞), but it does not change the solution set of (1) by the weak
maximum principle, since all the solution of (1) satisfies 0 ≤ u(x) ≤ c(x). Then the energy
functional Iλ(u) = Φ(u) − λΨ(u) : X → R associated with problem (1) is well defined.
Then it is easy to see that Iλ ∈ C1 (X,R) is weakly lower semi-continuous and u ∈ X is a
weak solution of (1) if and only if u is a critical point of Iλ. From the regularity assumptions
on f(x, u), any critical point u of Iλ(·) is a classical solution of (1) (see [19, 20]), and from
the strong maximum principle and the definition of modified f(x, u) above, u is either zero or
satisfies 0 < u(x) < c(x) for any x ∈ Ω. Moreover, we have

I ′λ(u)v = M

(∫
Ω

1

2
|∇u|2 dx

)∫
Ω
∇u∇v dx− λ

∫
Ω
f(x, u)v dx

= Φ′(u)v − λΨ′(u), for any v ∈ X.

From (M ) and Lemma 4.1 of [21] we can easily see that Φ′ is of (S+) type, i.e. if un ⇀ u in X
and lim

n→+∞
(Φ′(un)− Φ′(u), un − u) ≤ 0, then un → u in X . It is clear that Ψ′ is weak-strong

continuous (or see Lemma 1.2 of [1]). So I ′λ is of (S+) type.



Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 85

Our main existence result is as follows:

Theorem 2.1. If M(t) satisfies (M ) and f(x, u) satisfies (f1)–(f3), and Ω1 is an open subset
of Ω such that ∫ c(x)

0
f(x, s) ds > 0 (5)

for x ∈ Ω1, then for λ large enough, (1) has at least two positive solutions, and (1) has no
solution for small λ.

In order to prove our main existence result we need the following lemma:

Lemma 2.1 (see [1].) Suppose that f satisfies (f1)–(f3). If u(x) is an integrable function
in Ω, and there is a measurable subset Ω0 of Ω with positive measure, such that∫ c(x)

0
f(x, s) ds > 0 in Ω0 and

∫ c(x)

0
f(x, s) ds ≤ 0 in Ω \ Ω0,

then ∫ u(x)

0
f(x, s) ds ≤

∫ c(x)

0
f(x, s) ds in Ω0 and

∫ u(x)

0
f(x, s) ds ≤ 0 in Ω \ Ω0,

Now we turn to the nonexistence of the positive solutions of (1) when (5) does not hold
for any x ∈ Ω. We define c = maxx∈Ω c(x), f(u) = maxx∈Ω f(x, u). Our main nonexistence
result is

Theorem 2.2. If
∫ c

0 f(u) du ≤ 0, then (1) has no positive solution for any λ > 0.

In order to prove our main nonexistence result, we recall a theorem in [22] for (1) with
the special case of M(t) ≡ 1 and f(x, u) ≡ f(u). In fact, the theorem also holds for the
nonlocal problem (1) with f(x, u) ≡ f(u). Because the proof is similar to the proof of[22],
we omit it here (for detail, see the proof of Theorem 1 in [22]). Let us assume that f : R→ R is
a C1 function and let the following conditions hold: there exist 0 ≤ s0 < s1 < s2, such that

f(si) = 0, i = 1, 2,
f(s0) ≤ 0,
f(s) < 0, s0 < s < s1,
f(s) > 0, s1 < s < s2

(6)

and let ∫ s2

s0

f(s) ds ≤ 0. (7)

We have the following lemma.

Lemma 2.2. Assume that f satisfies (6) and (7). Let Ω be a bounded domain with smooth
boundary. If (1) with f(x, u) ≡ f(u) has a positive solution u, then u can not satisfy{

umax = maxx∈Ω u(x) ∈ (s1, s2),
u(x) > 0, x ∈ Ω.

(8)



86 Ma:Existence and Nonexistence of Positive Solutions for a Kirchhoff-type . . . . . .

Remark 2.2. Note that our assumptions (f1)–(f3) are weaker than (f1)–(f4) of[1] even in the
case ofM(t) ≡ 1. In fact, from (f1)–(f3), we can easily see that there exist a positive constant
β such that f(x, s) ≤ βs for any s ≥ 0 and a.e. x ∈ Ω, i.e., the condition (f4) of [1]. We do
not need the conditions of b(x) ∈ C1,α(Ω)(0 < α < 1) and f(·, u) ∈ C1,α(Ω) for any u ≥ 0
because we do not need energy functional of (1) is aC2 function in X in our proof.

Remark 2.3. The condition of f(x, ·) ∈ C1(R+) for any x ∈ Ω can be relaxed to f(x, ·)
is locally lipschitz in R+ for any x ∈ Ω. In fact, Lemma 2.2 also holds when f : R → R
is a locally lipschitz function because the symmetry results of [23] holds under this weaker
condition.

3. Proof of Theorem 2.1 and 2.2

In this section we will prove Theorem 2.1 and 2.2.

Lemma 3.1. If M(t) satisfies (M ), and f(x, u) satisfies (f1)–(f3) and (5), then for λ large
enough, Iλ(·) has a global minimum point u1 such that Iλ(u1) < 0.

Proof. Since
∫ c(x)

0 f(x, s) ds > 0 in Ω1, then there exists a measurable set Ω0 ⊂ Ω with

positive measure, such that
∫ c(x)

0 f(x, s) ds > 0 in Ω0, and
∫ c(x)

0 f(x, s) ds ≤ 0 in Ω \ Ω0.
From (M ) and the definition of M̂(t), we have M̂(t) ≥ m0t. In view of Lemma 2.1, we have

Iλ(u) = M̂

(∫
Ω

1

2
|∇u|2 dx

)
− λ

∫
Ω
F (x, u) dx

≥ m0

∫
Ω

1

2
|∇u|2 dx− λ

∫
Ω

(∫ u(x)

0
f(x, s) ds

)
dx

≥ m0

∫
Ω

1

2
|∇u|2 dx− λ

∫
Ω0

(∫ u(x)

0
f(x, s) ds

)
dx− λ

∫
Ω\Ω0

(∫ u(x)

0
f(x, s) ds

)
dx

≥ m0

∫
Ω

1

2
|∇u|2 dx− λ

∫
Ω0

(∫ c(x)

0
f(x, s) ds

)
dx

≥ m0

∫
Ω

1

2
|∇u|2 dx− λ

∫
Ω0

A1 dx

=
m0

2
‖u‖2 − λ|Ω0|A1 → +∞, as ‖u‖ → +∞, (9)

where A1 = maxx∈Ω0
|F (x, c(x))|. Since Iλ is weakly lower semi-continuous, Iλ has a mini-

mum point u1 in X .

Next we shall prove Iλ(u1) < 0, thus u1 is a positive solution of (1). In fact, we only need
to verify that when λ is large there exists a u0 ∈ X , such that Iλ(u0) < 0 = Iλ(0). We define
u0(x) = 0 in Ω \ Ω1ε, and u0(x) = c(x) in Ω1 and properly in Ω1ε \ Ω1 such that u0 ∈ X ,
where Ω1ε = {x ∈ Ω : dist(x,Ω1) ≤ ε}. Using the similar method with[1], we have

Iλ(u0) ≤ M̂
(∫

Ω

1

2
|∇u0|2 dx

)
− λ

∫
Ω1

F (x, c(x)) dx− λ [−A1 (|Ω1ε| − |Ω1|)] dx. (10)



Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 87

Since
∫ c(cx)

0 f(x, s) ds > 0 when x ∈ Ω1 and
∫ c(x)

0 f(x, s) ds is continuous, then there must

exists an open subset Ω2 with Ω2 ⊂ Ω1 and δ > 0, such that |Ω2| > 0 and
∫ c(x)

0 f(x, s) ds ≥ δ
for x ∈ Ω2. Choose ε small enough, such that δ|Ω2| + A(|Ω1| − |Ω1ε|) > 0. Again using the
similar method with [1], we have

Iλ(u0) ≤ M̂
(∫

Ω

1

2
|∇u0|2 dx

)
− λ [δ|Ω2|+A1(|Ω1| − |Ω1ε|)] .

Therefore when λ large enough, Iλ(u0) < 0, and consequently when λ is large enough, (1)
has a positive solution u1(x) satisfying Iλ(u1) = infu∈X Iλ(u) < 0.

Next we shall use Mountain Pass Theorem to prove that (1) has another positive solution
u2. First we prove Iλ(u) satisfies Palais-Smale condition.

Definition 3.1. We say that Iλ satisfies (P.S.) condition in X , if any sequence {un} ⊂ X
such that {Iλ(un)} is bounded and I ′λ(un) → 0 as n → +∞, has a convergent subsequence,
where (P.S.) means Palais-Smale.

Lemma 3.2. If M(t) satisfies (M ), f satisfies (f1)–(f3) and (5), then Iλ satisfies (P.S.)
condition.

Proof. Suppose that {un} ⊂ X , |Iλ(un)| ≤ c0 and I ′λ(un) → 0 as n → +∞. In view of
(9), we have

c0 ≥ Iλ(un) ≥ m0

2
‖un‖2 − λ|Ω0|A.

Hence, {‖un‖} is bounded. Without loss of generality, we assume that un ⇀ u, then I ′(un)(un−
u)→ 0. Therefore, we have un → u by the (S+) property of I ′λ.

Lemma 3.3. IfM(t) satisfies (M ), f satisfies (f1)–(f3), then there exist ρ > 0 and γ > 0 such
that Iλ(u) ≥ γ for every u ∈ X with ‖u‖ = ρ.

Proof. We define b = minx∈Ω. For any u(x) ∈ X , we also define B1 = {x ∈ Ω : u(x) < b},
B2 = {x ∈ Ω : u(x) ≥ b}. It is well known that the embedding of X ↪→ Lp(Ω) is continuous
when 2 < p ≤ 2∗, where 2∗ is the critical exponent. From poincaré inequality, we have

b|B2|
1
p ≤

(∫
B2

up dx

) 1
p

≤ c1

(∫
B2

|∇u|2 dx
) 1

2

≤ c1

(∫
Ω
|∇u|2 dx

) 1
2

= c1‖u‖,



88 Ma:Existence and Nonexistence of Positive Solutions for a Kirchhoff-type . . . . . .

where c1 is the embedding constant of X ↪→ Lp(Ω). Thus, we have

Iλ(u) = M̂

(∫
Ω

1

2
|∇u|2 dx

)
− λ

∫
Ω
F (x, u) dx

≥ m0

2
‖u‖2 − λ

∫
B1

F (x, u) dx− λ
∫
B2

F (x, u) dx

≥ m0

2
‖u‖2 − λ

∫
B2

F (x, u) dx

≥ m0

2
‖u‖2 − λA2|B2|

≥ m0

2
‖u‖2 − λA2

(
c1

b

)p
‖u‖p

= ‖u‖2
(
m0

2
− λA2

(
c1

b

)p
‖u‖p−2

)
,

where A2 = max(x,s)∈B2×[b,c] |F (x, s)|. Therefore, there exist m0b
p

2λA2c
p
1
> ρ > 0 such that

Iλ(u) ≥ ρ2
(
m0
2 − λA2

(
c1
b

)p
ρp−2

)
:= γ > 0 for every ‖u‖ = ρ and fixed λ.

Proof of Theorem 2.1 concluded. First let us show that Iλ satisfies the conditions of Mountain
Pass Theorem (see Theorem 2.10 of [24]). By Lemma 3.2, Iλ satisfies (P.S.) condition in X .
By Lemma 3.3, for fixed λ > 0, there exist min

{
‖u0‖, m0b

p

2λA2c
p
1

}
> ρ > 0, γ > 0 such that

Iλ(u) ≥ γ > 0 for every ‖u‖ = ρ, where u0 comes from (10). On the other hand, since
Iλ(0) = 0 and from the proof Lemma 3.1, there exists u0 ∈ X such that Iλ(u0) < 0 and
‖u0‖ > ρ. So from Mountain Pass Theorem, Iλ has another critical point u2 such that

Iλ(u2) ≥ γ > 0 > Iλ(u1).

Therefore, u2 is another positive solution of (1).

Finally we show that (1) has no positive solution when λ is small. We assume (1) has a
positive solution u, let (Λ1, ϕ1(x)) be the principal eigen-pair of the problem{

−∆φ = Λφ in Ω,
u = 0 on ∂Ω,

(11)

such that ϕ1(x) > 0 in Ω. We rewrite (1) as the following form{
−∆u = λ f(x,u)

M(
∫
Ω

1
2
|∇u|2 dx)

in Ω,

u = 0 on ∂Ω.
(12)

Multiplying (11) by u, multiplying (12) by ϕ1, subtracting and integrating in Ω, we obtain

0 =

∫
Ω

[
Λ1uϕ1 − λϕ1

f(x, u)

M (t)

]
dx =

∫
Ω

uϕ1

M (t)

[
M (t) Λ1 − λ

f(x, u)

u

]
dx, (13)

where t =
∫

Ω
1
2 |∇u|

2 dx. If λ < m0Λ1
β , then by Remark 2.2, we have

M (t) Λ1 − λ
f(x, u)

u
≥ m0Λ1 − λ

f(x, u)

u
> m0Λ1 − λβ > 0.



Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 89

That is contrary to (13). So for small λ, (1) has no positive solution.
Proof of Theorem 2.2. The proof is similar to the proof of [1]. For the sake of completeness,
we include it here. If there exists a positive solution (λ, u∗) for (1), then u∗ is a subsolution
of {

M
(∫

Ω
1
2 |∇u|

2 dx
)

∆u+ λf(u) = 0 in Ω,
u = 0 on ∂Ω,

(14)

since M
(∫

Ω
1
2 |∇u∗|

2 dx
)

∆u∗ + λf(u∗) ≥ M
(∫

Ω
1
2 |∇u∗|

2 dx
)

∆u∗ + λf(x, u∗). And c is
supersolution of (13). So by the standard comparison arguments, (13) has a positive solution u
such that u∗ ≤ u ≤ c. But if we let s0 = 0, s1 = b and s2 = c, f satisfies (6) and (7), then
by Lemma 2.2, (13) has no positive solution. This is a contradiction. So (1) has no positive
solution if

∫ c
0 f(u) du ≤ 0.

4. Proof of a Conjecture and Some Examples

In this section we will prove the conjecture of Liu, Wang and Shi’s and give some typical
consequences of Theorem 2.1 to Theorem 2.2.

In [1], Liu, Wang and Shi conjecture that the nonexistence holds with a weaker condition:∫ c(x)

0
f(x, s) ds ≤ 0 for any x ∈ Ω. (15)

In fact, as we will see in the following Proposition, the condition (15) is more strong than∫ c
0 f(s) ds ≤ 0. Therefore, by Theorem 2.2, the conjecture is right.

Proposition 4.1. If f(x, u) satisfies (f1)–(f3) and
∫ c(x)

0 f(x, s) ds ≤ 0 for any x ∈ Ω, we
have

∫ c
0 f(s) ds ≤ 0.

Proof. From (f1)–(f3), we can easily see that f(x, s) ≤ 0 when s ∈ [c(x), c]. Thus, we
have

∫ c
c(x) f(x, s) ds ≤ 0. Then, for any x ∈ Ω, we have

0 ≥
∫ c(x)

0
f(x, s) ds =

∫ c

0
f(x, s) ds−

∫ c

c(x)
f(x, s) ds ≥

∫ c

0
f(x, s) ds.

In particular,
∫ c

0 f(s) ds ≤ 0.

Now, we give some examples which satisfy our hypotheses.

Example 4.1. Let M(t) = a + bt with t =
∫

Ω
1
2 |∇u|

2 dx, here a, b are two positive con-
stants and f(x, u) = u(u − b(x))(c(x) − u) with b(x) ∈ C(Ω), c(x) ∈ C1(Ω) such that
0 < b(x) < c(x) for any x ∈ Ω. It is clear that M(t) and f(x, u) verify our assumptions (M)
and (f1)–(f3).

Example 4.2. We consider a special case of Example 4.1:{
∆u+ λu(u− b(x))(c(x)− u) = 0 in Ω,
u = 0 on ∂Ω,

(16)



90 Ma:Existence and Nonexistence of Positive Solutions for a Kirchhoff-type . . . . . .

where b(x) ∈ C(Ω), c(x) ∈ C1(Ω) such that 0 < b(x) < c(x) for any x ∈ Ω. We have known
that f(x, u) satisfies (f1)–(f3) from Example 4.1. Moreover, we have∫ c(x)

0
f(x, s) ds =

∫ c(x)

0
s(s− b(x))(c(x)− s) ds

=
1

12
[c(x)]3(c(x)− 2b(x)).

Then by Theorem 2.1, if there exists an open subset Ω1 ⊂ Ω, such that c(x) > 2b(x) in Ω1,
then (16) has at least two positive solutions for large λ.

If c(x) ≡ 1 for all x ∈ Ω, we obtain∫ 1

0
f(s) ds =

∫ 1

0
max
x∈Ω

s(s− b(x))(1− s) ds

=

∫ 1

0
max
x∈Ω

[s2 − s3 + b(x)(s2 − s)] ds

=
1

12
− b

6
,

since s2 − s ≤ 0 for s ∈ [0, 1]. Then by Theorem 2.2, if b = minx∈Ω b(x) ≥ 1
2 , then (16) has

no positive solution for any λ > 0.

Example 4.3. Let M(t) ≡ 1 and f(x, s) = s(s − 1)(c(x) − s) with 3
2 ≤ c(x) for any

x ∈ Ω. We can easily obtain∫ c

0
f(s) ds =

∫ c

0
max
x∈Ω

s(s− 1)(c(x)− s) ds

=

∫ c

0
(c(x)s2 − s3 + s2 − c(x)s) ds

=
c3

3
− c4

4
+

∫ c

0
max
x∈Ω

c(x)(s2 − s) ds

=
c3

3
− c4

4
+ max

x∈Ω
c(x)

(
c3

3
− c2

2

)
ds

=
c3

3
− c4

4
+ c

(
c3

3
− c2

2

)
ds

=
c3

12
[c− 2].

So
∫ c

0 f(s) ds ≤ 0 if and only if c ≤ 2.

On the other hand,∫ c(x)

0
f(x, s) ds =

∫ c

0
s(s− 1)(c(x)− s) ds−

∫ c

c(x)
s(s− 1)(c(x)− s) ds

≥
∫ c

0
s(s− 1)(c(x)− s) ds

= −c
4

4
+

1 + c(x)

3
c3 − c(x)

2
c2.



Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 91

If
∫ c(x)

0 f(x, s) ds ≤ 0 for any x ∈ Ω, we have

0 ≥ −c
4

4
+

1 + c(x)

3
c3 − c(x)

2
c2

⇒ 4(1 + c(x))c− 6c(x) ≤ 3c2.

In particular, we have
4(1 + c)c− 6c ≤ 3c2 ⇒ c ≤ 2.

However, it is clear that∫ c

0
f(s) ds ≤ 0 ;

∫ c(x)

0
f(x, s) ds ≤ 0 for any x ∈ Ω.

Therefore, the condition “
∫ c(x)

0 f(x, s) ds ≤ 0 for any x ∈ Ω” is more strong than the condition
“
∫ c

0 f(s) ds ≤ 0” in this example, which verifies Proposition 4.1 by a concrete example.

Remark 4.1. In [25], Dancer and Yan proved when c(x) ≡ 1 and {x ∈ Ω : b(x) < 1/2}
is of positive measure, then (16) may have many positive solutions of local minimum type.
The results of Example 4.2 shows that the condition

∫ 1
0 f(s) ds ≤ 0 is optimal for the nonexis-

tence of positive solution of (16). However, we do not know whether
∫ c

0 f(s) ds ≤ 0 is optimal
for the nonexistence of positive solution of (1).

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